The Weak Lefschetz Property for monomial complete intersections

Abstract

Let $A=k[x_1,...,x_n]/(x_1^d,...,x_n^d)$, where $k$ is an infinite field. If $k$ has characteristic zero, then Stanley proved that $A$ has the Weak Lefschetz Property (WLP). Henceforth, $k$ has positive characteristic $p$. If $n=3$, then Brenner and Kaid have identified all $d$, as a function of $p$, for which $A$ has the WLP. In the present paper, the analogous project is carried out for $4\le n$. If $4\le n$ and $p=2$, then $A$ has the WLP if and only if $d=1$. If $n=4$ and $p$ is odd, then we prove that $A$ has the WLP if and only if $d=kq+r$ for integers $k,q,d$ with $1\le k\le \frac{p-1}{2}$, $r\in \frac{q-1}{2},\frac{q+1}{2}$, and $q=p^e$ for some non-negative integer $e$. If $5\le n$, then we prove that $A$ has the WLP if and only if $\lfloor \frac{n(d-1)+3}{2}\rfloor \le p$. We first interpret the WLP for the ring $k[x_1,...,x_n]/(x_1^d,...,x_n^d)$ in terms of the degrees of the non-Koszul relations on the elements $x_1^d,...,x_{n-1}^d,(x_1+...+x_{n-1}{)}^d$ in the polynomial ring $k[x_1,...,x_{n-1}]$. We then exhibit a sufficient condition for $k[x_1,...,x_n]/(x_1^d,...,x_n^d)$ to have the WLP. This condition is expressed in terms of the non-vanishing in $k$ of determinants of various Toeplitz matrices of binomial coefficients. Frobenius techniques are used to produce relations of low degree on $x_1^d$, ..., $x_{n-1}^d$, $(x_1+...+x_{n-1}{)}^d$. From this we obtain a necessary condition for $A$ to have the WLP. We prove that the necessary condition is sufficient by showing that the relevant determinants are non-zero in $k$.

Cited by in corpus (3)

• The Lefschetz properties of monomial complete intersections in positive characteristic
• The weak Lefschetz property for Artinian Gorenstein algebras of codimension three
• On the degrees of relations on $x_1^{d_1},\dots ,x_n^{d_n},(x_1+\dots +x_n{)}^{d_{n+1}}$ in positive characteristic